# Information Partition and Common Knowledge

Person $$A$$ has information partition $$\{\{1,2\},\{3,4\},\{5,6\}\}$$

Person $$B$$ has information partition $$\{\{1\},\{2,3\},\{4,5\},\{6\}\}$$

$$w=3$$ has been realized.

I'm confused about what $$A$$ knows that $$B$$ knows.

At first I thought of it as $$K_aK_b(3)=K_a(\{2,3\})=\{1,2,3,4\}$$, but the example I found online says $$K_aK_b(3)=\{2,3,4,5\}$$. why?

The situation is not described by $$K_A(K_B(3))$$. $$A$$ does not know that the draw was 3, and hence does not know that $$B$$ thinks it is in the set $$K_B(3)$$. What $$A$$ does know is that the draw is in set $$K_A(3)$$, hence $$A$$ thinks $$B$$ should have narrowed it down to $$K_B(K_A(3))$$.